After four of the frostiest years in over six decades of U.S.-Israel relations, the two leaders are taking pains to send friendlier signals — in part because Obama will be around for another four years, in part because centrist and left-leaning parties in Israel unexpectedly elbowed their way into Netanyahu’s hawkish coalition during elections in January.

“For this, you need, you see, a second term as president and a third term as prime minister. That really fixes things,” Netanyahu said in response to a question about Obama’s standing in Israel.

The two men seemed to go out of their way to appear chummy Wednesday at a press conference, trading jokes about their children’s good looks, and working hard to seem aligned no matter the question or the issue that came up.

Repeatedly, Netanyahu tried to reassure a skittish Israeli public that the president is more of an ally than many of them think, batting away the skepticism at one point with a simple, “People should get to know President Obama the way I’ve gotten to know him.”

There’s no missing the change.

“Netanyahu has been able to get a bump every time he says no to the president. When he upbraided him in the Oval Office, he got a 10 percent rise in the Israeli public opinion. That is unheard of,” said Martin Indyk, Bill Clinton’s ambassador to Israel, during a roundtable discussion in Washington last week.

“Netanyahu, [who] eats polls for breakfast, he knows very well what the standing of the president is and what his own standing is,” said Indyk, who predicted “a more pliant Netanyahu” more open to Obama’s suggestions on settlements and Palestine.

The result, Indyk said, is a “very careful, very sensitive, very praiseworthy” Netanyahu which will help the prime minister’s poll numbers because “the public doesn’t like the idea that their prime minister doesn’t have a good relationship with the president.”

In the push-and-pull of their relationship, Obama has less incentive to give ground. Appetite for another armed conflict in the Mideast is low, and resentments linger. But the president is still eager to stave off an Israeli military strike that could unleash violence around the region. The only way to do that may be to convince Netanyahu to trust him.

”Obama’s got the most dysfunctional relationship with any Israeli prime minister in the history of the U.S.-Israeli relationship,” former State Department official Aaron David Miller said on MSNBC Monday. “Benjamin Netanyahu bears an enormous amount of responsibility for that.”

“It was a bad four years from a personality point of view. Now, they know they’ve got to work together for another four years,” says Elliott Abrams, a deputy national security under President George W. Bush. “How about trying a fresh start? From the point of view of both governments, that’s very sensible [and] timely.”

It isn’t only about fresh starts, it’s about a rapidly shifting political landscape in Israel. The Israeli elections augur a generational shift towards politicians of Obama’s age or younger, if not always the president’s progressive point of view.

The 64-year-old Israeli prime minister lost serious ground to Naftali Bennett, the right-wing, 40-year-old, American-born leader of the Jewish Home party and Yair Lapid, the telegenic 49-year-old former TV news anchor who heads the moderate Yesh Atid party, forming an ungainly coalition government days before Obama arrived.

“Bibi is not the same Bibi he dealt with over the last four years,” Miller told POLITICO. “Two forty-somethings have changed the channel in Israel….You’ve got Bennett. You‘ve got Lapid now.”

Obama may have underscored the generational issue when he told Israeli security officials at the airport arrival ceremony Wednesday that he’d rather walk across the tarmac.

“I’m a young man. I’m always looking for any chance to walk,” the 51-year-old Obama said after hopping out of an armored SUV.

A short time later, Obama stripped off his suit coat and slung it over his shoulder. Netanyahu quickly took the cue and did the same, creating an early iconic image of the trip.

In recent weeks, Netanyahu has toned down his criticism of Obama’s Iran policy, and his government had nothing negative to say, at least publicly, when Vice President Joe Biden announced the U.S. would resume so-called P5+1 negotiations with Tehran shortly after Obama’s second-term swearing-in.

Obama set a lighter tone during an appearance last week on Israeli television, referring to Netanyahu by his nickname “Bibi” no less than four times, and joking about donning a fake mustache so he could sip coffee with college students, unnoticed.

Netanyahu picked up on that Wednesday, joking that he’d made arrangements.

“I took note of your desire to go incognito around Israel, so if you have a few free minutes, and you can arrange to slip away from your security – a daunting task – well, we picked out a few cafes and bars in Tel Aviv, and we even prepared a fake mustache for you,” the prime minister said, to laughter from Obama.

Obama joined in the somewhat awkward banter. After being told at the Tel Aviv airport to follow a red line on the ground, he joked about Netanyahu’s repeated warnings over the Iranian nuclear program: “He’s always talking to me about red lines.”

Going into the trip, Obama sounded less than effusive about his dealings with the Israeli leader.

”We’ve got a terrific, businesslike relationship. He is very blunt with me about his views on issues and I’m very blunt with him about my views on issues and we get stuff done,” Obama told Israel’s Channel 2. “There have been times that Bibi and I have had differences but, as I said, the relationship between the two countries is so strong between the people is so strong I think that any difference in policy, not personal, but policy differences end up being bridged and resolved.”

Netanyahu on Wednesday also spoke about their dealings in business terms. “I want to thank you for the investment you have made in our relationship…It is deeply, deeply appreciated,” he said at the press conference.

For their part, Israelis remain skeptical, even hostile, to a president who made a landmark trip to Cairo four years before stepping foot in the Jewish State — and some local commentators argued, acidly, that his trip comes too soon after the formation of a new government and too close to the start of Passover, a frenzied time of shopping and cupboard-clearing for Israelis.

Obama’s “Israel sojourn will bring together two leaders struggling to hide their respective convictions that they know better how to safeguard not only their own country but also their counterpart’s,” wrote liberal columnist David Horovitz in the English-language Times of Israel. “ There’s no doubt that the alliance between the two countries is solid, deeply rooted, based on a range of genuine common values and interests, and particularly important given the Middle East’s current instability and unpredictability. Unfortunately, there’s also no doubt that each of these two particular leaders believes the other is disastrously wrong-headed. And if that gulf between them is not bridged, it could have phenomenally damaging consequences.”

The president has described the 51-hour trip to the Jewish State as nothing more committal than a listening tour, as if it were one of his pre-election bus tours of purple-state America. He ultimately hopes to jump-start the Israel-Palestinian peace process, and rues Netanyahu’s decision to proceed with settlement construction on the West Bank, but he’s not going to ram it down anyone’s throat, according to administration officials.

Obama remains much less popular than other recent U.S. presidents, according to recent polls, although Israeli public opinion seems especially volatile when it comes to No. 44. A Jerusalem Post poll conducted in the days before Obama’s visit found that 36 percent of Israelis thought Obama was pro-Palestinian, 26 percent viewed him as pro-Israel and a quarter judged him somewhere in the middle.

That was actually Obama’s best showing in four years. His “pro-Israel” rating bottomed out at 4 percent in 2009, after he clashed with Netanyahu over Iran and other issues.

During the 2012 campaign, many in Obama’s orbit viewed Netanyahu — who speaks college professor English with the whisper of a Philadelphia accent — as a meddlesome outsider not-so-secretly rooting for Mitt Romney, who had promised a more aggressive policy towards Israel’s archenemy Iran.

“I stand with our friends in Israel. I stand with our allies,” Romney said during a September rally in Florida — after reports surfaced claiming Obama snubbed a meeting with Netanyahu for an appearance on David Letterman’s show. “I can’t ever imagine, if the prime minister of Israel asked to meet with me, I can’t ever imagine saying no. … They’re our friends, they’re our closest allies in the Middle East.”

“Israeli meddling in internal U.S. affairs and turning the U.S. administration from an ally to ‘an enemy’ has caused us severe damage,” said Shaul Mofaz, leader of the centrist opposition Kadima party, at the time. “Please explain to us: who is Israel’s greatest enemy — the U.S. or Iran? Who do you fear more — [Iranian President Mahmoud] Ahmadinejad or Obama? Which regime is more important to overthrow — the one in Washington, or in Tehran?”

By mid-September, Netanyahu had concluded that Romney would lose — and shifted to a strategy of engaging Obama on friendlier terms, a former U.S. intelligence official who huddled with top Likud security officials told POLITICO at the time.

Obama’s big win, coupled with Netanyahu’s tempered victory, could prompt a new phase of collaboration between the two leaders. But longtime observers say Israel’s domestic political situation is too roiled for any real meeting of the minds and warn that any policy shifts will likely be determined by events in Tehran, Cairo or Damascus — not by this week’s trip.

“Every time Netanyahu starts a new government or some kind of new thing happens, we talk about now he’s seen the light and there will be something new and he will move and he wants his legacy,” said Natan Sachs, a fellow at the Saban Center for Middle East Policy at Brookings. “The best guide for Netanyahu’s foreign policy in the next few years, as long as this coalition lasts, is his foreign policy in his first two terms, which was, shall we say, not in line with the way President Obama sees the Palestinian issue. “